在三角形ABC中,a+c=2b,角A-角C=60度,则SinB=_____
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解决时间 2021-08-12 22:35
- 提问者网友：辞取
- 2021-08-11 22:13
在三角形ABC中,a+c=2b,角A-角C=60度,则SinB=_____
最佳答案
- 五星知识达人网友：玩家
- 2021-08-11 23:21
则由正弦定理，得：a=2RsinA,b=2RsinB,c=2RsinC,代入a+c=2b，得： 2RsinA+2RsinC=4RsinB,即sinA+sinC=2sinB
2sin[(A+C)/2] * cos[(A-C)/2] = 2 * 2 * sin(B/2) * cos(B/2)
sin[(п-B)/2] * cos30 = 2 * sin(B/2) * cos(B/2)
cos[B/2] * cos30 = 2 * sin(B/2) * cos(B/2)
因为0度<B/2<90度，所以cos(B/2)非零
cos30 = 2 * sin(B/2)
sin(B/2) = 根3 / 4
则cos(B/2)=√（1-sin^2(B/2)=√13/4
则sinB=2sin(B/2)cos(B/2)=√39/8
2sin[(A+C)/2] * cos[(A-C)/2] = 2 * 2 * sin(B/2) * cos(B/2)
sin[(п-B)/2] * cos30 = 2 * sin(B/2) * cos(B/2)
cos[B/2] * cos30 = 2 * sin(B/2) * cos(B/2)
因为0度<B/2<90度，所以cos(B/2)非零
cos30 = 2 * sin(B/2)
sin(B/2) = 根3 / 4
则cos(B/2)=√（1-sin^2(B/2)=√13/4
则sinB=2sin(B/2)cos(B/2)=√39/8
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